On the generalized Stirling formula - Creative Math. and Inf.

CREATIVE MATH. & INF.19 (2010), No. 1, 53 - 56Online version available at http://creative-mathematics.ubm.ro/Print Edition: ISSN 1584 - 286X Online Edition: ISSN 1843 - 441XOn the generalized Stirling formulaCRISTINEL MORTICIABSTRACT.The aim of this paper is to give a numerical construction of the best approximations of the factorial function among the family of approximationsintroduced by Mortici [Arch. Math. (Basel) 93 (2009), No. 1, 37-45].1. INTRODUCTIONThere are many situations when for solving some practical problems, big factorials must be estimated. Probablythe most known estimation formula for the factorial function is the Stirling’s formula:n! ∼ √ ( n) n2πn(1.1)ewhich will be written for some reasons in the equivalent form:n! ∼ √ ( n) n+ 122πee= σ n . (1.2)The formula (1.1) was discovered by the French mathematician Abraham de Moivre (1667-1754) in the form( n) n √n! ∼ k ne(where the constant was not explicitly given) when he was preoccupied to give the normal approximation for thebinomial distribution.Afterwards, the Scottish mathematician James Stirling (1692-1770) discovered the constant √ 2π, using the Wallis’formula. Both used a formula of type Euler-MacLaurin to estimate the sum ln 2 + ln 3 + ... + ln n. For proofs anddetails see for example [13]-[15].A slightly more accurate estimate is the Burnside formula [1]:n! ∼ √ 2πand other better formulas were found [2], but a sacrifice of simplicity.Recently, Mortici [3] introduced the formula( n +1)n+ 1 22= β n (1.3)eand the more general family of approximations√2πn! ∼e( n + 1e) n+ 12= αn( ) n+b n + an! ∼ λ= µ n (λ, a, b), (1.4)ewhere the constants λ, a, b will be determined to obtain performant results. Please see also [3]-[12] for other details.Interesting fact is that also the Stirling’s formula is a particular case of (1.4), obtained for λ = √ 2πe, a = 0 and b = 1/2,as we can see from the representation (1.2).In order to have approximations of type (1.4), the basic condition that the sequence (µ n (λ, a, b)/n!) n≥1convergesto 1 must be fulled. It is obtained in [3] the following final form of approximations, for x ∈ [0, 1] ,n! ∼ √ ( ) n+ 1n + x2πe · e −x 2= µn (x),eand then the best constants x ∈ {ω, ζ} , ω = ( 3 − √ 3 ) /6 and ζ = ( 3 + √ 3 ) /6 are deduced to provide the bestapproximations. These constants were found considering some completely monotonic functions.In this paper we refind these constants using a different numerical method.Received: 23.09.2009; In revised form: 18.01.2010; Accepted: 08.02.2010.1991 Mathematics Subject Classification. 40A25.Key words and phrases. Factorial function, gamma function, Stirling’s formula, Burnside’s formula.53

54 C. Mortici2. THE RESULTSThe function µ n (x) is strictly increasing on [0, 1/2] and strictly decreasing on [1/2, 1] , becauseddx (ln µ n(x)) =12 − xx + n ,so a direct consequence is that µ n (0) < n! < µ n (1/2) > n! > µ n (1), orσ n < n! < β n > n! > α n . (2.5)By the inequalities (2.5) and by continuity arguments on the function µ n (x), it results that there exists a sequence(w n ) n≥1⊂ (0, 1/2) such that µ n (w n ) = n!, which isn! = √ ( ) n+ 1n + wn22πe · e −wn eand there exists a sequence (z n ) n≥1⊂ (1/2, 1) such that f(z n ) = n!, orn! = √ ( ) n+ 1n + zn22πe · e −zn .eAfter some numerical computations, we deduced that the sequences (w n ) n≥1and (z n ) n≥1are decreasing, so they areconvergent to w = 0.21133..., respective to z = 0.78868... . Now we have obtained the following under-approximationand upper-approximation formulas for the factorial function:√2πe · e−0.21133...( n + 0.21133...e) n+ 12< n!

On the generalized Stirling formula 55REFERENCES[1] Burnside, W., A rapidly convergent series for log N!, Messenger Math. 46 (1917), 157-159[2] Gosper, R. W., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. 75 (1978), 40–42[3] Mortici, C., An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. 93 (2009), No. 1, 37-45[4] Mortici, C., Product approximations via asymptotic integration, Amer. Math. Monthly 117 (2010),No. 5, 434-441[5] Mortici, C., New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett. 23 (2010), No. 1, 97-100[6] Mortici, C., New sharp bounds for gamma and digamma functions, An. Ştiinţ. Univ. A. I. Cuza Iaşi Ser. N. Matem. 56, No. 2 (in press)[7] Mortici, C., Complete monotonic functions associated with gamma function and applications, Carpathian J. Math. 25 (2009), No. 2, 186-191[8] Mortici, C., Optimizing the rate of convergence in some new classes of sequences convergent to Euler’s constant, Anal. Appl. (Singap.) 8 (2010), No. 1,1-9[9] Mortici, C., Improved convergence towards generalized Euler-Mascheroni constant, Appl. Math. Comput. 215 (2010), 3443-3448[10] Mortici, C., A class of integral approximations for the factorial function, Comput. Math. Appl. (2010), DOI: 10.1016/j.camwa.2009.12.010[11] Mortici, C., Best estimates of the generalized Stirling formula, Appl. Math. Comput. 215 (2010), No. 11, 4044-4048[12] Mortici, C., On new sequences converging towards the Euler-Mascheroni constant, Comput. Math. Appl. (2010), DOI: 10.1016/j.camwa.2010.01.029[13] O’Connor, J. and Robertson, E. F., James Stirling, MacTutor History of Mathematics Archive[14] Qin, X. and Su, Y.,Viscosity approximation methods for nonexpansive mappings in Banach spaces, Carpathian J. Math. 22 (2006), No. 1-2, 163-172[15] Stirling, J., Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium, London, 1730. English translation by J. Holliday,The Differential Method: A Treatise of the Summation and Interpolation of Infinite SeriesDEPARTMENT OF MATHEMATICSVALAHIA UNIVERSITY OF TÂRGOVIŞTEUNIRII 18, 130082 TÂRGOVIŞTEE-mail address: cmortici@valahia.roE-mail address: cristinelmortici@yahoo.com